Nuprl Lemma : fps-compose-add
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[g,f,h:PowerSeries(X;r)].
((g+h)(x:=f) = (g(x:=f)+h(x:=f)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-add: (f+g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-compose: g(x:=f),
fps-add: (f+g),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
all: ∀x:A. B[x],
implies: P ⇒ Q,
crng: CRng,
comm: Comm(T;op),
and: P ∧ Q,
cand: A c∧ B,
rng: Rng,
listp: A List+,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
true: True,
infix_ap: x f y,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv)
Lemmas referenced :
bag-parts'_wf,
bag_wf,
listp_wf,
rng_plus_comm,
crng_properties,
rng_all_properties,
listp_properties,
bag-product_wf,
rng_car_wf,
rng_times_wf,
rng_one_wf,
tl_wf,
list-subtype-bag,
equal_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
rng_plus_wf,
rng_zero_wf,
bag-append_wf,
hd_wf,
bag-rep_wf,
length_wf_nat,
bag-summation_wf,
infix_ap_wf,
bag-summation-add,
rng_properties,
squash_wf,
true_wf,
assoc_wf,
comm_wf,
rng_times_over_plus,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
independent_isectElimination,
hypothesis,
lambdaFormation,
setElimination,
rename,
productElimination,
because_Cache,
independent_pairFormation,
applyEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
universeEquality,
natural_numberEquality,
imageElimination,
productEquality,
functionExtensionality,
functionEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[g,f,h:PowerSeries(X;r)].
((g+h)(x:=f) = (g(x:=f)+h(x:=f)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_36
Last ObjectModification:
2017_07_26-PM-06_33_57
Theory : power!series
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